EigenvalueResults

  1. Recall the eigenvalue decomposition: If A = V D V-1, where the diagonal entries of the diagonal matrix D are the eigenvalues of A and the columns of V are the eigenvectors of A.
  2. Defn: A is similar to B if A = S B S-1 for a non-singular matrix S.
  3. Thm: Similar matrices have the same eigenvalues.
  4. Thm: If A is symmetric all its eigenvalues are real.
  5. Thm: If A is symmetric A = V D V-1 = V D VT, where D is diagonal and V is orthogonal. Therefore the eigenvectors of a symmetric matrix are orthogonal.
  6. Thm: The eigenvectors of a (real) nonsymmetric matrix are real or come in complex conjugate pairs. Ex: has eigenvalues and .
  7. Defn: A (real) matrix is normal if AT A = A AT. Note: A symmetric matrix is normal but a normal matrix may not be symmetric. Ex: is not symmetric but is normal.
  8. Defn: A complex matrix is normal if AHA = A AH where AH = complex conjugate of AT.
  9. Defn: Q is orthogonalif Q is a real matrix and if QT Q = I.
  10. Defn: Q is unitary if QH Q = I. Note: An orthogonal matrix is nitary but a unitary matrix may not be orthogonal. Ex: is unitary but not orthogonal. (This Q happens to also be normal.)
  11. Thm: If A is normal, then A = V D V-1 = V D VH where D is diagonal and V is unitary. Ex: for the normal matrix, A = V D VH with , .
  12. Defn:An n by n matrix is defective if it does not have n linearly independent eigenvectors. In this case the eigenvalue decomposition A = V D V-1 with D diagonal and V nonsingular is not possible. Ex: is defective.
  13. A matrix that is not defective is called diagonalizable. In this case the eigenvalue decomposition A = V D V-1 with D diagonal and V nonsingular always exists.
  14. Thms: If n by n matrix A has n distinct eigenvalues then it is diagonalizable. If A is normal then it is diagonalizable. If A is symmetric then it is diagonalizable.
  15. Remark: If A is defective one can decompose A using Jordan form. (However note that Jordan form is not a numerically stable decomposition.)
  16. Thm: For any matrix A, A can be decomposed into its Schur decomposition A = V T V-1 = V T VH where V is unitary and T is triangular with diag(T) = eigenvalues of A. (This is numerically stable.)
  17. Accuracy of eigenvalues: Given a matrix and a nearby matrix letbe an eigenvalue of A and be the eigenvalue of that is closest to . Assume that A is diagonalizable so that A = V D V-1 is possible. Thm:
  18. Remark: If A is symmetric or more generally if A is normal then cond(V) = 1 and eigenvalues can be calculated accurately.
  19. Remark: If A is far from a symmetric matrix or more generally far from a normal matrix then cond(V) can be large. In this case eigenvalues are difficult to calculate accurately. Ex: is far from a symmetric matrix. For this matrix the eigenvalue decomposition has and so that the eigenvalues are 1 and 2. Here is large and so small inaccuracies in A will lead to large changes in the corresponding eigenvalues. For example is a small change in A and whereas the eigenvalues of are 0.38 and 2.26 which are not close to the eigenvalues of A.
  20. Remark: The set of eigenvalues of A is called the spectrum of A. For matrices that are far from normal matrices the concept of psuedospectra is important (see )
  21. Thm. If Av =v and AHw =conj() wfor unit vectors v, wthen . So get larger errors if w (eigenvector of AH) and v (eigenvector of A) are almost orthogonal.